Hall's relations in finite groups.
type: Item
instance of: scholarly article
title: Hall's relations in finite groups.
zbMATH Open document ID: 1044.20008
author: Masafumi Murai
author: Yugen Takegahara
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
full work available at URL: https://muroran-it.repo.nii.ac.jp/
review text: Let \(G\) be a finite group, let \(n\) be a fixed positive integer and let \(\alpha\in\Aut(G)\) be such that the order of \(\alpha\) divides \(n\). Let \(l_n(G,\alpha)\) denote the number of elements of \(G\) satisfying the equality \(xx^\alpha x^{\alpha^2}\cdots x^{\alpha^{n-1}}=1\). \textit{Ph. Hall} [Proc. Lond. Math. Soc., II. Ser. 40, 468-480 (1935), 481-501 (1936; Zbl 0015.20201)] extended a well-known result of Frobenius by showing that \((n,|G|)\) divides \(l_n(G,\alpha)\). In the article under review, the authors show that certain arithmetic properties of the number \(l_n(G,\alpha)\) have a strong impact on the internal structure of \(G\). Specifically, the main result asserts that if the prime \(p\) divides \((n,|G|)\) and if \((pn,|G|)\) does not divide \(l_n(G,\alpha)\), then a Sylow \(p\)-subgroup of \(G\) is either cyclic (when \(p\) is odd), or is exceptional (when \(p=2\)). Here ``exceptional'' means cyclic, dihedral, generalized quaternion, or quasi-dihedral.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 2037094
zbMATH Keywords: automorphisms
zbMATH Keywords: Sylow subgroups
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
OpenAlex ID: W2054978893
cites work: \(|\Hom(A,G)|\). IV
cites work: \(|\Hom(A,G)|\). II.
cites work: Q5848209
cites work: Q4124997
cites work: A Contribution to the Theory of Groups of Prime-Power Order
cites work: On a conjecture of Frobenius
cites work: Q4312077
cites work: Über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung in \(p\)-Gruppen
cites work: Artin exponent of finite groups
cites work: On the number of \(p\)-subgroups of a finite group
DOI: 10.1016/J.JALGEBRA.2003.06.010
description: scientific article; zbMATH DE number 2037094